Theorem isphg 31106

Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, and the norm is 𝑁. An inner product space is also called a pre-Hilbert space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isphg.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isphg	⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem isphg

Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ph 31102	. . 3 ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})
2	1	elin2 4164	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}))
3		rneq 5924	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
4		isphg.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
5	3, 4	eqtr4di 2822	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
6		oveq 7414	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
7	6	fveq2d 6883	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
8	7	oveq1d 7423	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦))↑2) = ((𝑛‘(𝑥𝐺𝑦))↑2))
9		oveq 7414	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(-1𝑠𝑦)) = (𝑥𝐺(-1𝑠𝑦)))
10	9	fveq2d 6883	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑠𝑦))))
11	10	oveq1d 7423	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2))
12	8, 11	oveq12d 7426	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)))
13	12	eqeq1d 2771	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
14	5, 13	raleqbidv 3345	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
15	5, 14	raleqbidv 3345	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
16		oveq 7414	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (-1𝑠𝑦) = (-1𝑆𝑦))
17	16	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑥𝐺(-1𝑠𝑦)) = (𝑥𝐺(-1𝑆𝑦)))
18	17	fveq2d 6883	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑛‘(𝑥𝐺(-1𝑠𝑦))) = (𝑛‘(𝑥𝐺(-1𝑆𝑦))))
19	18	oveq1d 7423	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2) = ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2))
20	19	oveq2d 7424	. . . . . 6 ⊢ (𝑠 = 𝑆 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
21	20	eqeq1d 2771	. . . . 5 ⊢ (𝑠 = 𝑆 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
22	21	2ralbidv 3235	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))))
23		fveq1 6878	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
24	23	oveq1d 7423	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝑥𝐺𝑦))↑2))
25		fveq1 6878	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝑥𝐺(-1𝑆𝑦))))
26	25	oveq1d 7423	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2))
27	24, 26	oveq12d 7426	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)))
28		fveq1 6878	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
29	28	oveq1d 7423	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥)↑2) = ((𝑁‘𝑥)↑2))
30		fveq1 6878	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
31	30	oveq1d 7423	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑦)↑2) = ((𝑁‘𝑦)↑2))
32	29, 31	oveq12d 7426	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)) = (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))
33	32	oveq2d 7424	. . . . . 6 ⊢ (𝑛 = 𝑁 → (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
34	27, 33	eqeq12d 2785	. . . . 5 ⊢ (𝑛 = 𝑁 → ((((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
35	34	2ralbidv 3235	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑛‘(𝑥𝐺𝑦))↑2) + ((𝑛‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
36	15, 22, 35	eloprabg 7518	. . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
37	36	anbi2d 641	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → ((⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
38	2, 37	bitrid 286	1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵 ∧ 𝑁 ∈ 𝐶) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⟨cop 4597  ran crn 5660  ‘cfv 6534  (class class class)co 7408  {coprab 7409  1c1 11097   + caddc 11099   · cmul 11101  -cneg 11438  2c2 12291  ↑cexp 14093  NrmCVeccnv 30873  CPreHil_OLDccphlo 31101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6490  df-fv 6542  df-ov 7411  df-oprab 7412  df-ph 31102

This theorem is referenced by:  cncph  31108  isph  31111  phpar  31113  hhph  31467